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Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena. (English) Zbl 1072.35098

The paper deals with the Cauchy–Dirichlet problem for one-dimensional semilinear parabolic equations with a gradient nonlinearity of the form \[ \left\{\begin{align*}{& u_t=u_{xx}+|u_x|^p,\quad t>0,\ 0<x<1,\cr & u(t,0)=0,\quad u(t,1)=M,\qquad t>0,\cr & u(0,x)=u_0(x),\qquad 0<x<1,\cr }\end{align*}\right. \] where \(p>2,\) \(M\geq0\) and \(u_0\in X:=\{v\in C^1[0,1]\colon\;v(0)=0,\;v(1)=M\}.\)
The authors provide a complete classification of large time behaviour of the classical solutions \(u.\) Precisely, either the space derivative \(u_x\) blows up in a finite time with \(u\) itself remaining bounded, or \(u\) is global and converges in \(C^1\)-norm to the unique steady state. The main difficulty concerns the proof of \(C^1\)-boundedness of all global solutions, and in avoinding it the authors compute a nontrivial Lyapunov functional by carrying out a method introduced by T. Zelenyak. After deriving precise estimates on the solutions and on the Lyapunov functional, they argue by contradiction by showing that any \(C^1\) unbounded global solution should converge to a singular stationary solution, which does not exist.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K65 Degenerate parabolic equations
35B45 A priori estimates in context of PDEs
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